Yes, the formula above is correct. Well, it depends on what we mean by correct.
NDVI does not make sense
Imagine the following situation. We have fetched a cloud-free mosaic of Sentinel 2 satellite data and want to measure NDVI (Normalised difference vegetation index), which uses red and near-infrared bands within this simple formula.
The results are normalised, which in this case means that they lie between -1 and 1. Always.
We open out raster data using xarray and have all 4 bands in a single xarray.DataSet. The code to measure NDVI is then simple.
>>> red = data.sel(band=1) # select red band
>>> nir = data.sel(band=4) # select near-infrared band
>>> ndvi = (nir - red) / (nir + red) # compute NDVI
And, a surprise! Our results are between 0 and 170. That is certainly not correct. What has happened?
16-bit unsigned integer
The data coming from Sentinel 2 are stored as 16-bit unsigned integer (uint16). That means that the value the array can hold can be anything between 0 and 216 – 1 (65,535). Remember that NDVI is between -1 and 1. Does it mean that uint16 cannot represent NDVI values? Yes, precisely.
Look at this toy example to understand what is happening during the computation of NDVI in uint16. Let’s have an array with four numbers and subtract 10 from each of them.
Yes, as weird as it is, it is correct. The result should be negative, but we can’t have negative values in uint16. So what happens is that the counter rolls over and subtracts the remaining value from the maximum it can represent (65,535 – x).
It is exactly like a rollover of the odometer. We ran out of values, so they started over. The only difference is that we have 16 binary values encoding each number, not decimal.
The fix is easy. We have to use data type which does not limit you like this, like a 64-bit integer.
This is a short story of one open-source algorithm and its journey from QGIS to mapclassify, to be used within GeoPandas. I am writing it to illustrate the flow within the open-source community because even though this happens all the time, we normally don’t talk about it. And we should.
Sometimes last year, I asked myself a question. How hard would it be to port topological colouring tool from QGIS to be used with GeoPandas? Considering that this particular QGIS tool is written in Python, it seemed to be an easy task.
For those of you who never used it, the aim of topological colouring is to assign colours to (usually) polygons in such a way, that no two adjacent polygons share the same colour (see the illustration in the embedded tweet below).
The adaptation of the Nyall Dawson’s original algorithm was quite straightforward, the logic of Python algorithms for QGIS and for GeoPandas is the same. So in October, I have asked the others what would be the ideal way of sharing it.
The original license was not compatible with the one we use in GeoPandas and I was not sure if GeoPandas itself is actually the right place for it. So while thinking about it, Nyall himself made the situation easier and offered to relicense the original code.
However, there was no clear consensus what is the best way at that time and the whole idea was set aside, until the end of the year, when I decided to keep it simple and release the script as a tiny Python package. And greedy was born.
In the end, greedy offered the original QGIS algorithm for topological colouring and a few other options on top of that, applicable to all geometry types (unlike QGIS, which is limited to polygons). It was a simple solution to release it like that, but it was not optimal, because the more scattered the ecosystem is, the more complicated is to navigate in it and people just easily miss things.
We could have ended the story here, but then Levi Wolf came with an idea.
After some discussion, splot was not seen as the best place, but mapclassify was. And after a couple of months, I made a pull request merging greedy into mapclassify.
It is a very simple story, you may say. Yes, but it shows one thing very clearly, and that is a willingness of the community to collaborate across different projects. A lot of people were involved in it, everyone willing to find the best solution. I think it is worth sharing tiny stories like this.
To see the capability of mapclassify.greedy, here is a Jupyter notebook for you. Thanks everyone involved!
This is just a quick appendix to the story, outlining the translation of the code from QGIS to GeoPandas-compatible version.
First thing to say – it is easy! Easier that expected, to be honest. I might have been lucky with the algorithm I’ve chosen, but I believe that there is a scope for other processing tools to be ported this way.
The code of greedy is here and original code here.
QGIS code has a lot of stuff related to QGIS interface, which can be happily ignored. The core is in def processAlgorithm(self, parameters, context, feedback) and that is the part we should focus on.
Nyall’s code can be roughly divided into three steps (ignoring some interface code):
Compute graph to understand topological relationships between polygons
Compute balanced topological colouring
Assign colours to features
To compute graph, Nyall defines a new class holding the topology and checks which polygons intersect with which. I did not want to have a specific class, because we have libpysal‘s weights object taking care of it. Moreover, it comes with an alternative option of topology inferring as contiguity weights. No need to expensively compute intersections anymore (unless we want to, I kept the option there).
Balanced colouring is, in the end, the only part of the code, which is almost entirely original. I made only a few modifications.
Because topological colouring is a know Graph problem, there is a selection of algorithms in networkx library dealing with the problem, so I just simply linked them in.
Finally, the function now returns pandas.Series, making it simple to assign resulting values to GeoDataFrame. The most simple usage is then a single line.
Sometimes our lines and polygons are way too complicated for the purpose. Let’s say that we have a beautiful shape of Europe, and we want to make an interactive online map using that shape. Soon we’ll figure out that the polygon has too many points, it takes ages to load, it consumes a lot of memory and, in the end, we don’t even see the full detail. To make things easier, we decide to simplify my polygon.
Simplification means that we want to express the same geometry, using fewer points, but trying to preserve the original shape as much as we can. The easiest way is to open QGIS and use its Simplify processing tool. Now we face the choice – which simplification method should we use? Douglas-Peucker or Visvalingam? How do they work? What is the difference? What does a “tolerance” mean?
This short post aims to answer these questions. I’ll try to explain both of these most popular algorithms, so you can make proper decisions while using them.
First let’s see both how algorithms simplify the following line.
Douglas-Peucker, or sometimes Ramer–Douglas–Peucker algorithm, is the better known of the two. Its main aim is to identify those points, which are less important for the overall shape of the line and remove them. It does not generate any new point.
The algorithm accepts typically one parameter, tolerance, sometimes called epsilon. To explain how is epsilon used, it is the best to start with the principle. Douglas-Peucker is an iterative algorithm – it removes the point, splits the line and starts again until there is no point which could be removed. In the first step, it makes a line between the first and the last points of the line, as illustrated in the figure below. Then it identifies the point on the line, which is the furthest from this line connecting endpoints. If the distance between the line and the point is less than epsilon, the point is discarded, and the algorithm starts again until there is no point between endpoints.
If the distance between the point and the line is larger than epsilon, the first and the furthest points are connected with another line and every point, which is closer than epsilon to this line gets discarded. Every time a new furthest point is identified, our original line splits in two and the algorithm continues on each part separately. The animation below shows the whole procedure of simplification of the line above using the Douglas-Peucker algorithm.
Visvalingam-Whyatt shares the aim with Douglas-Peucker – identify points which could be removed. However, the principle is different. Tolerance, or epsilon, in this case, is an area, not a distance.
Visvalingam-Whyatt, in the first step, generates triangles between points, as illustrated in the figure below.
Then it identifies the smallest of these triangles and checks if its area is smaller or larger than the epsilon. If it is smaller, the point associated with the triangle gets discarded, and we start again – generate new triangles, identify the smallest one, check and repeat. The algorithm stops when all generated triangles are larger than the epsilon. See the whole simplification process below.
A great explanation of Visvalingam-Whyatt algorithm with an interactive visualisation made Mike Bostock.
Which one is better?
You can see from the example above, that the final line is the same, but that is not always true, and both algorithms can result in different geometries. Visvalingam-Whyatt tends to produce nicer geometry and is often preferred for simplification of natural features. Douglas-Peucker tends to produce spiky lines at specific configurations. You can compare the actual behaviour of both at this great example by Michael Barry.
Which one is faster?
Let’s figure it out. I will use a long randomised line and Python package simplification, which implements both algorithms. The results may vary based on the actual implementation, but using the same package seems fair. I generate randomised line based on 5000 points and then simplify if using both algorithms with the epsilon fine-tuned to return a similar number of points.
import numpy as np
from simplification.cutil import (
simplify_coords, # this is Douglas-Peucker
simplify_coords_vw, # this is Visvalingam-Whyatt
# generate coords of 5000 ordered points as a line
coords = np.sort(np.random.rand(5000, 2), axis=0)
# how many coordinates returns DP with eps=0.01?
# 30 / 5000
# how many coordinates returns VW with eps=0.001?
# 28 / 500
And the winner is – Douglas-Peucker. By a significant margin.
74.1 µs ± 1.46 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
2.17 ms ± 23.9 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
Douglas-Peucker is clearly more performant, but Visvalingam-Whyatt can produce nicer-looking geometry, pick the one you prefer.
Percentage instead of epsilon
Some implementations of simplification algorithms do not offer tolerance / epsilon parameter, but ask for a percentage. How many points do you want to keep? One example of this approach is mapshaper by Matthew Bloch. Based on the iterative nature of both, you can figure out how that works :).
What about topology?
It may happen, that the algorithm (any of them) returns invalid self-intersecting line. Be aware that it may happen. Some implementations (like GEOS used by Shapely and GeoPandas) provide optional slower version preserving topology, but some don’t, so be careful.
I have gaps between my polygons
If you are trying to simplify GeoDataFrame or shapefile, you may be surprised that the simplification makes gaps between the polygons where there should not be any. The reason for that is simple – the algorithm simplifies each polygon separately, so you will easily get something like this.
If you want nice simplification which preserves topology between all polygons, like mapshaper does, look for TopoJSON. Without explaining how that works, as it deserves its own post, see the example below for yourself as the last bit of this text.
import topojson as tp
topo = tp.Topology(df, prequantize=False)
If there’s something inaccurate or confusing, let me know.